Modulation, Transmission and Radio Communication

1Why voices need a carrier

Speech and music lie between about 20 Hz and 20 kHz. Could a radio station simply feed its microphone signal to an aerial? At 20 kHz the wavelength is $\frac{3.00\times {10}^8}{2.0\times {10}^4}=15\backslash frac\{3.00\; \backslash times\; 10^\{8\}\}\{2.0\; \backslash times\; 10^\{4\}\}\; =\; 15$ km, and an efficient aerial must be a sizeable fraction of a wavelength, kilometres long. Worse, every station would broadcast in the same frequency band, and a receiver could not separate them. The solution is modulation: the audio signal is made to vary some property of a high-frequency carrier wave, and each station is given its own carrier frequency. An FM station at 100 MHz needs an aerial only about 1.5 m long, and dozens of stations share the air without interfering.

2Amplitude modulation (AM)

In AM, the amplitude of the carrier is made to follow the audio signal. For a carrier $V_c\sin (2\pi f_{c}t)V\_c\backslash sin(2\backslash pi\; f\_c\; t)$ and an audio signal $V_m\sin (2\pi f_{m}t)V\_m\backslash sin(2\backslash pi\; f\_m\; t)$,

The ratio $m=\frac{V_m}{V_c}m\; =\; \backslash frac\{V\_m\}\{V\_c\}$ is the modulation index. It must not exceed 1, or the envelope folds over and the audio is distorted (over-modulation).

Using $\sin A\sin B=\frac{1}{2}[\cos (A-B)-\cos (A+B)]\backslash sin\; A\backslash sin\; B\; =\; \backslash tfrac\{1\}\{2\}\backslash left[\backslash cos(A\; -\; B)\; -\; \backslash cos(A\; +\; B)\backslash right]$, the AM wave is the sum of three sine waves:

Besides the carrier there are two side frequencies, $f_c-f_{m}f\_c\; -\; f\_m$ and $f_c+f_{m}f\_c\; +\; f\_m$, each of amplitude $\frac{m{V}_c}{2}\backslash frac\{mV\_c\}\{2\}$. Real audio contains a range of frequencies up to $f_{\max }f\_\{\backslash max\}$, which produces two sidebands, and the bandwidth of the AM signal is

Since power is proportional to amplitude squared, the total power is $P_c(1+\frac{m^2}{2})P\_c\backslash left(1\; +\; \backslash frac\{m^2\}\{2\}\backslash right)$, where $P_{c}P\_c$ is the carrier power. All the information is in the sidebands; the carrier itself carries none.

3Frequency modulation (FM)

In FM the amplitude of the carrier stays constant and its frequency is made to vary with the audio signal: louder sound swings the frequency further from $f_{c}f\_c$, and higher-pitched sound swings it more often. The greatest change of frequency is the frequency deviation $\mathrm{\Delta }f\backslash Delta\; f$. The bandwidth needed is given approximately by Carson's rule:

Broadcast FM uses $\mathrm{\Delta }f=75\backslash Delta\; f\; =\; 75$ kHz and audio up to 15 kHz, so $B\approx 2(75+15)=180B\; \backslash approx\; 2(75\; +\; 15)\; =\; 180$ kHz; stations in the 87.5–108 MHz band are spaced 200 kHz apart. AM broadcasting in Africa and Europe uses channels only 9 kHz apart.

Digital modulation

Digital data is carried by switching a property of the carrier between fixed values: its amplitude (ASK, amplitude-shift keying), its frequency (FSK) or its phase (PSK). Wi-Fi and 4G combine amplitude and phase steps (QAM), so that each symbol carries several bits at once.

4How radio waves travel

• Ground (surface) wave, below about 3 MHz (long and medium wave): the wave follows the curve of the Earth for hundreds of kilometres, losing energy to the ground.
• Sky wave, about 3–30 MHz (short wave): the wave is refracted back to Earth by the ionosphere, a layer of ionised gas 60–400 km up, and can hop round the world.
• Space wave, above about 30 MHz (VHF FM, TV, mobile phones): the wave passes through the ionosphere, so a receiver must be within line of sight of the transmitter. That is why FM and TV transmitters are put on hilltops and tall masts.
• Satellite links use microwaves, which pass straight through the ionosphere to satellites and back.

For an aerial at height $hh$ above flat ground on an Earth of radius $RR$, the distance to the horizon is found from Pythagoras: $(R+h{)}^2=R^2+d^2(R\; +\; h)^2\; =\; R^2\; +\; d^2$, and since $h\ll Rh\; \backslash ll\; R$,

5The radio receiver

Nearly every radio is a superheterodyne receiver. The mixer combines the incoming signal with a local oscillator whose frequency is set by the tuning dial, always a fixed amount above the wanted carrier. This shifts the chosen station to a fixed intermediate frequency (IF), where most of the amplification and filtering is done by circuits that never need retuning. The demodulator then recovers the audio.